Take 1/2^n common

(1/2)^n[4007^n + 4009^n]

(1/2)^n[(4008-1)^n + (4008+1)^n]

(1/2)^n-1 [4008^n + nC2x4008^n-2.... even terms

Now this holds for n=1, n=3 but not for n=5.. Because there you get a term 4008x5.. Which has only 3 powers of 2 in it while the denominator has 4..

So answer is n=1,3